Nonlinear reconstruction can yield large improvements over linear reconstruction when the sensors or features are carefully selected [1]}.
Successful nonlinear reconstruction techniques include neural networks [2]},[3]}, deep nonlinear state estimators [4]}, [5]}, and convex \(\ell ^1\)  minimization to reveal sparse coefficients in learned libraries [6]}, [7]}.
The need for nonlinear representation and reconstruction is also recognized in the reduced-order-modeling community where it is called “nonlinear Galerkin” approximation [8]}, [9]}, [10]}.
These methods are necessary because in many systems of interest, the state is found to lie near a low-dimensional underlying manifold that is curved in such a way that it is not contained in any low-dimensional subspace [11]}.
We will show that the best possible linear reconstruction accuracy is fundamentally limited by the number of measurements (features) and the fraction of the variance that is captured in the principal subspace [12]} of that dimension.
In essence, any linear representation in a subspace is “too loose” and demands an excessive number of measurements to even have a hope of accurately reconstructing the state using linear functions.
Nonlinear reconstruction is much more powerful, as Whitney's celebrated embedding theorem (Theorem. 5, [13]}) shows that states on any \(r\) -dimensional smooth manifold can be reconstructed using \(2 r\)  carefully chosen measurements.
If the measurements must be linear functions of the state on a compact sub-manifold of \(\mathbb {R}^n\)  then \(2r+1\)  can be found [14]}.
